Examples in Action
3. Concrete Examples to Solidify Understanding
Let's solidify this with some more examples. Pretend you're baking a cake. The ingredients are your elements, and the topology is the recipe. Change the recipe, and you get a different cake (or maybe a disaster!).
Example 1: The Real Number Line with the Usual TopologyLet's take the singleton {3.14}. The complement of {3.14} is all real numbers except 3.14. You can draw an open interval around any of those numbers that doesn't include 3.14. Therefore, the complement is open, making {3.14} closed.
Example 2: A Finite Set with the Discrete TopologyLet's say our set is {A, B, C}, and we use the discrete topology. In this case, {A} is both open and closed because every subset is open. So easy, it almost feels like cheating!
Example 3: The Real Number Line with the Lower Limit TopologyThis one's a bit trickier. In the lower limit topology, intervals of the form [a, b) are open. So, is {5} closed? Well, its complement is (-, 5) (5, ). The interval (5, ) can be written as a union of [5+1/n, 5+1/(n+1)) as n goes to infinity, which makes it open in lower limit topology. However, (-, 5) cannot be written as a union of intervals [a,b) as it would need to include 5. Thus, {5} is closed here too.
So, When Is a Singleton Closed?
4. A Summary and Key Takeaways
Alright, let's recap. We've seen that the answer to "Is a singleton always closed?" is a resounding "It depends!" The crucial factor is the topology you're working with. In the usual topology on the real numbers, yes, singletons are closed. In the discrete topology, absolutely! But in other, more exotic topologies (like the indiscrete topology we looked at earlier), it might not be the case.
The key takeaway is that 'closed' and 'open' are relative concepts. They're defined by the structure imposed by the topology. It's not an inherent property of the set itself, but rather its relationship to the other sets in the space, as defined by the topological structure. Don't get hung up on the individual set; think about the whole neighborhood it lives in.
Thinking about the complement is crucial. If you can show that the complement of your singleton is an open set according to the topology's rules, then you've proven that the singleton is closed. If you can't, then it's not. It's like a detective game where you're trying to fit the pieces together according to the rules of the case.
In conclusion, the question isn't about whether singletons are closed, but when they are. Explore different topologies, play around with examples, and you'll start to develop an intuition for when a singleton will behave nicely and be closed, and when it might decide to be a rebel and not conform. Remember, math is all about exploring and understanding the underlying structures.
